Girsanov theorem

Girsanov Theorem is a fundamental result in the field of probability theory and stochastic processes, particularly in the study of Brownian motion and martingales. It provides a method for changing the probability measure on a space of paths in a way that transforms a Brownian motion under the original measure into a Brownian motion with drift under the new measure. This theorem has significant applications in mathematical finance, especially in the pricing of financial derivatives, and in risk management.

Statement of the Theorem

Let \( (\Omega, \mathcal{F}, \mathbb{P}) \) be a probability space, and let \( \mathbb{Q} \) be another probability measure on \( \Omega \) that is absolutely continuous with respect to \( \mathbb{P} \) (denoted \( \mathbb{Q} \ll \mathbb{P} \)). Let \( \{W_t\}_{t \geq 0} \) be a Brownian motion under \( \mathbb{P} \), and let \( \{\mathcal{F}_t\}_{t \geq 0} \) be the filtration generated by \( W_t \). Suppose there exists an \( \mathcal{F}_t \)-adapted process \( \{\theta_t\}_{t \geq 0} \) satisfying certain integrability conditions. Then, under the measure \( \mathbb{Q} \), defined by the Radon-Nikodym derivative

\[ \frac{d\mathbb{Q}}{d\mathbb{P}} \bigg|_{\mathcal{F}_t} = \exp\left( \int_0^t \theta_s dW_s - \frac{1}{2} \int_0^t \theta_s^2 ds \right), \]

the process

\[ \tilde{W}_t = W_t - \int_0^t \theta_s ds \]

is a Brownian motion with respect to \( \mathbb{Q} \).

Applications

The Girsanov Theorem is crucial in the field of quantitative finance, where it is used to derive the Black-Scholes equation for option pricing. By changing the measure from the "real world" probability measure to the "risk-neutral" measure, one can simplify the problem of pricing derivatives by removing the drift of the underlying asset's price process. This allows for the valuation of derivatives to be based solely on the risk-free rate, irrespective of the asset's expected return.

Proof

The proof of Girsanov's Theorem involves verifying that \( \tilde{W}_t \) satisfies the definition of a Brownian motion under the measure \( \mathbb{Q} \). This includes showing that \( \tilde{W}_t \) has independent and stationary increments, and that for any \( t \), \( \tilde{W}_t \) is normally distributed with mean 0 and variance \( t \). The proof also relies on the properties of the stochastic exponential and the concept of martingales.

Limitations and Conditions

The application of Girsanov's Theorem requires the process \( \theta_t \) to satisfy Novikov's condition or a similar condition ensuring the exponential martingale is a true martingale. These conditions are necessary to ensure the absolute continuity of measures and the integrability of the Radon-Nikodym derivative.

See Also

• Martingale (probability theory)
• Brownian motion
• Radon-Nikodym theorem
• Stochastic calculus
• Mathematical finance